I missed this.
First, I commend you for taking in these dogs when they had nowhere else to go. It's a difficult situation.
This can be corrected, but it will take some work, and one of the solutions (which the trainer is likely going to stress) is something you'll have difficulty with, which is exercise. Lots of exercise. You have three dogs of two breeds from strong working lines. They need to work, it's instinctual. They had no other job, really, so they made one up themselves (which is very common in these situations), and they take that job very seriously; they also feed off eachother. In order for them to be really satisfied, i.e. happy and content, with a new job, it has to be one they like better. Just trying to squash their instincts of the job they have now will lead to their replacement with another job, and likely other behavior problems. In dog language. All that will lead to is a houseful of unhappy beings.
So back to the exercise. (And I know you said you can't do it and can't afford it, but I'm going to explain why it's an important piece of your puzzle.) A tired dog is a happy dog. No truer words were ever spoken. A dog bred to work has energy it HAS to get out. It just has to. Expecting a working dog to not do much of anything all day and then not find something to do on his own is just not realistic. They will find something to do. They have to. Border Collies and Labs especially. (And I work with strong working breeds also, and have for a lifetime, so have seen it both ways - trained and exercised, and not.) So dogs go out and alas! It's work time! People to chase and bark at, yeah, that's the ticket!
Anyone would have a hard time getting them to calm down because their pent up energy is finally able to be released, and it's become learned and ingrained behavior up until recently. If you want to read about it google "fence running". It's the same type thing.
A pp had a great suggestion of trying to find someone who could help exercise your dogs. I think you should really take another look at that suggestion. Look at it this way. I'm sure you're going to spend quite a bit of money on a trainer. Maybe that same money could've been used instead to hire a reliable kid in the neighborhood to take your dogs for a daily (or even 3 or 4 day weekly) run and play with them. Do you have a dog park near where you live? Here's why. When your dogs come back from a hard run, or hard playtime, they're pooped. It's a beautiful quiet that sometimes is so relaxing to the dog, you wonder if they're ok, lol. When I do that with my dog now, he's sacked until the next day practically, and he's young and full of energy. So I implore you to consider starting there. Somehow.
Next, you need to have control over the dogs when they are out in the yard. I think letting them out separately is a good idea for several reasons. It gives you more control and it takes away the part of them feeding off echother (like a bar room brawl when dogs are together - one starts off and the others jump in, then no way of stopping it, etc.) I would also find an inside room where the other dogs couldn't see the dog outside if you can't crate them. Take one dog out on a long leash. Must be leashed. I buy the 16 feet leads that are woven, not the type on a reel - they get dirty but you have more control over the leash > the dog. Work with him (others have given great suggestions so I won't repeat them) to reward the dog when he doesn't bark uncontrollably. (I would say a regular bark is ok, but not an excessive, out of control bark at people or things he sees outside the fence.) I use, "That's enough". It means, it's ok to bark, but not excessively. My dogs get it, and I'm sure yours can, too. If the dog continues to bark excessively, reel him in, put him in a sit, and say That's enough. (Or whatever term you want to use.) Make it happen. If he does it again, repeat. Over and over. At first, they will resist, of course. You have to show them you mean business. Never let them out without this leash and never let them go back to how it is now or you will have to start all over again. So I do think you have your work cut out for you. BUT, if you add in the exercise piece, it will be much easier than trying to deal with three hyper dogs who still have tons of energy to expend, kwim?
I am not a fan of bark - let's call them what they are - SHOCK collars. Unless it is in a very specific situation with someone who knows what they're doing AND is working on training the dog at the same time. The people I've seen use them are of the lazy training variety - don't want to put any effort into training and are looking for an easy solution. My last bite was from a friend's dog who was wearing one of these collars. I went to touch his neck and he bit me. The dog was only around a year old, and was already ruined. Needless to say, the dog was given away somewhere, who knows where. My friend didn't make one effort to train the dog, just took someone's stupid advice to put a shock collar on him. I know someone else who used one and when the dog didn't do what they wanted, they turned it up all the way until the dog was incontinent of urine and couldn't move. They thought it was kind of funny. I lost respect for all these people. Having a well behaved dog takes time and patience, and love and training. There are no easy solutions, really. If you (general you) can't put the time in, maybe you shouldn't have a dog. It's a living being and suffers from mistreatment and misunderstanding, etc. (Off soapbox.)
OP, I wish you luck. I feel for you because I can imagine how difficult this situation is for you. Do you know anyone who may be able to offer a home to any of the dogs? Even taking one out of the mix would help make the situation easier to manage. The reason I say this is because a situation like this can take a toll on your health. It's an ongoing, relentless problem, especially now with angry neighbors and the police involved. That can do a number on you if you have to deal with it constantly and can't get it under control. If it does get to the point of making you sick, then you might not be able to care for any of the dogs, so it might be kinder to think of rehoming one or two in this particular instance.
At least if the cops are called you can show proof that you are really TRYING to remedy the situation, which will hopefully keep your dogs in your care. It is probably going to take some time for all 3 dogs to get the barking to a minimum, but with your love and guidance I have a feeling it will all work out. Dogs are typically "pleasers" so hopefully they find the rewards for no barking more appealing than barking at kids or other dogs.
